byVincenzo Ferone; Carlo Nitsch; Cristina Trombetti

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We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar\'e inequalities. The key point is the study of an optimal weighted Wirtinger inequality. Source: http://arxiv.org/abs/1207.0680v2

It was recently proved that the elastic energy $E(\gamma)=\tfrac{1}{2}\int_\gamma\kappa^2 ds$ of a closed curve $\gamma$ with curvature $\kappa$ has a minimizer among all plane, simple, regular and closed curves of given enclosed area $A(\gamma)$, and that the minimum is attained only for circles. Here we show under which hypothesis the result can be extended to other functionals involving the curvature. As an example we show that the optimal shape remains a circle for the $p$-elastic energy... Topics: Optimization and Control, Analysis of PDEs, Mathematics Source: http://arxiv.org/abs/1606.01569

We show that the elastic energy $E(\gamma)$ of a closed curve $\gamma$ has a minimizer among all plane simple regular closed curves of given enclosed area $A(\gamma)$, and that the minimum is attained for a circle. The proof is of a geometric nature and deforms parts of $\gamma$ in a finite number of steps to construct some related convex sets with smaller energy. Topics: Mathematics, Optimization and Control Source: http://arxiv.org/abs/1411.6100

byVincenzo Ferone; Carlo Nitsch; Cristina Trombetti

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Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume. Topics: Optimization and Control, Differential Geometry, Mathematics Source: http://arxiv.org/abs/1604.06042

byAndrea Cianchi; Vincenzo Ferone; Carlo Nitsch; Cristina Trombetti

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Balls are shown to have the smallest optimal constant, among all admissible Euclidean domains, in Poincar\'e type boundary trace inequalities for functions of bounded variation with vanishing median or mean value. Source: http://arxiv.org/abs/1301.5770v1

byAndrea Cianchi; Vincenzo Ferone; Carlo Nitsch; Cristina Trombetti

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Extremal functions are exhibited in Poincar\'e trace inequalities for functions of bounded variation in the unit ball ${\mathbb B}^n$ of the $n$-dimensional Euclidean space ${\mathbb R}^n$. Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of ${\mathbb B}^n$, instead of just on $\partial {\mathbb B}^n$, as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the... Topics: Optimization and Control, Mathematics Source: http://arxiv.org/abs/1604.01524